In this section we will describe the construction of the wave function whis is used to solve the Lieb-Liniger equation in presence of an external confinement (refer to Sec. 3.2.5). The aim is to obtain a wave function suitable for the description of the gas in a wide range of the density, starting from the Tonks-Girardeau and up to the Gross-Pitaevskii regimes. The important point is that TG wave function always has nodes, although in GP regime the nodes are absent.
At the distances the interaction potential is absent and the solutions are
are simple sinus and cosine functions. We want to choose a solution which goes to
one at the matching distance , which is treated as a variational parameter,
and is a solution of a two-body problem (1.65) at a smaller distances. Also we
want to have a symmetry in sign reversing. The solution that satisfies those
conditions is
At the matching points the derivative is automatically equal to zero and
the function matches smoothly to a constant. The phase
is
related to the scattering length by the boundary condition
(1.70), which we will write as2.2
Once is obtained by solving numerically this equation the drift force
contribution (2.39) can be calculated from formula
(2.53) |
The energy contribution (refEE) is then described by
(2.54) |